Computer-assisted holographic method and device

ABSTRACT

The method consists of computing (E 1 -E 4 ) a set of two-dimensional images in a three-dimensional geometrical space representing a virtual object from respective different viewpoints. The virtual object is represented by digital data stored in a computer. Holograms are computed (E 5 -E 6 ) for the respective two-dimensional images using a fast complex transform, such as a Fourier transform. The holograms are then juxtaposed (E 7 ) to form a digital hologram of the object which is physically reproduced (E 8 ) by a spatial light modulator. A three-dimensional image of the object is obtained by illuminating the spatial light modulator with a coherent source.

The present invention relates generally to the reproduction ofthree-dimensional images and more particularly to the digital synthesisof holograms and the reproduction of three-dimensional images from theholograms.

Several techniques for reproducing three-dimensional images are known inthe art.

Some systems, referred to as “stereoscopic” systems, produce two offsetimages of the same scene, each of which is viewed by a respective eye ofan observer. Depth of field and volume are then reconstructed by thebrain of the observer. The majority of such systems require the observerto wear bulky accessories such as spectacles or headsets to separate anddifferentiate the images sent to each eye. Also, they have only abinocular angle of vision of the scene to be represented.

One particular example of another technique, referred to as the“autostereoscopic” technique, entails taking a photograph of a scenethrough a plane mesh of sufficiently small lenses so as to associate oneviewpoint of the photographed scene with each lens. The photographobtained in this way gives an illusion of relief but produces a limitedeffect of depth. This technique does not allow for the naturalaccommodation of the eye and in the current state of the art canreproduce three-dimensional images in real time only with difficulty.

Holograms are the most reliable way of reproducing three-dimensionalimages since they reproduce the optical field as generated by the scene.In particular, this method allows completely for the accommodation ofthe eye. Analog holograms are produced by projecting a coherent opticalwave emitted by a laser onto an object, picking up an optical wavediffused by the object and derived from that wave, and causing thediffused optical wave to interfere with a reference wave consisting ofanother portion of the beam emitted by the laser, to produce aninterference field which is recorded on a photosensitive medium such asa photographic plate. A three-dimensional image of the original scenecan then be observed by illuminating the photographic plate with acoherent wave. This purely analog method offers excellent reproductionquality but cannot reproduce three-dimensional images in real time.

There are digital holographic methods for producing three-dimensionalimages in real time. U.S. Pat. No. 5,668,648 describes acomputer-assisted holographic system capable of digitally synthesizingthe hologram of a virtual object and reproducing an image from thathologram. The virtual object is sampled into sampling points which aretreated as elementary spherical optical sources. Diffraction fields arecomputed for the respective sampling points and are then superposed. Aninterpolation technique is used to improve the resolution of theresulting diffraction field. An interference field (hologram) based onthe resultant diffraction field and data representing a reference waveis then generated and reproduced physically by a spatial lightmodulator.

The digital synthesis of holograms by the above method necessitates longand complex calculations, in particular to determine a diffraction fieldassociated with each sampling point on the object and to interpolate theresulting diffraction field.

The present invention aims to provide a holographic synthesis methodcapable of generating holograms digitally in real time, generallynecessitating less computation than the prior art methods, and providingthree-dimensional spatial reproduction of images.

To this end, the present invention provides a method of producing ahologram from a virtual object defined in a three-dimensionalgeometrical space, characterized in that it includes the followingsteps:

computing a set of two-dimensional images representing the object fromrespective different viewpoints in the three-dimensional space,

computing holograms respectively corresponding to said two-dimensionalimages, and

combining said holograms to form a hologram of the object.

The expression “virtual object” means data representing a real objectwhich is, for example, stored in a computer.

The steps defined above constitute a digital model of the production ofan analog hologram of a three-dimensional object. An analog hologram ismade up of a combination of elementary holograms, or diffractive fields,which reproduce two-dimensional images representing the object concernedfrom respective different viewpoints when they are individuallyilluminated by a coherent beam of light.

The step of computing the set of two-dimensional images advantageouslyincludes the following steps:

defining in the three-dimensional space a matrix of points in a firstgeometrical plane separate from the object, and

projecting images of the object as seen respectively from said points ofsaid matrix onto a second geometrical plane which is preferably betweenthe object and the first plane and parallel to the first plane, theprojected images constituting said two-dimensional images.

For each point of the matrix, the projection step preferably consists ofprojecting points of the object onto the second plane along respectivestraight lines passing through said points of the object and said eachpoint of the matrix.

According to the invention, the two-dimensional images are defined byrespective real functions and the step of computing the hologramscomprises the following steps for a given two-dimensional image:

transforming the given two-dimensional image defined by thecorresponding real function into a complex two-dimensional image definedby a complex function,

oversampling the complex image,

simulating the production of a diffracted image resulting from thediffraction of an optical wave by the oversampled complex image,

adding a complex field representing a reference optical wave to theresulting diffracted image, and

encoding values taken by the amplitude of the sum of said complex fieldand the resulting diffracted image to produce the hologram associatedwith the given two-dimensional image.

A “real or complex function” means a function of two variables takingreal or complex values, respectively. The real function is typically anintensity distribution and the complex function a distribution ofcomplex numbers each defined by a real amplitude and a real phase.

The step of transforming the given two-dimensional image into a compleximage derives from the given two-dimensional image an image which isdefined by complex numbers which are the optimum representation of theactual optical field and facilitate the computations employed in thestep of simulating the production of a diffracted image.

The step of oversampling the complex image increases the number ofpixels of the resulting hologram of the given two-dimensional imagebecause the computations employed during the subsequent simulation andaddition steps are performed on a greater number of image points. Theoversampling step can consist of inserting the complex image into alarger image in which the intensity of pixels outside the originalcomplex image is made equal to 0. In this case, implementing the step ofoversampling the complex image after the steps of transforming thetwo-dimensional image into a complex image avoids the need to computethe complex function for points of the oversampled image outside theoriginal complex image.

The transform step can include the following steps:

determining amplitude values each depending on the square root of acorresponding value taken by the real function of the giventwo-dimensional image, and

associating a phase with each of the amplitude values so that anamplitude value and a phase value are defined for each point of thecomplex image.

By averaging the amplitude values of the hologram, associating a phasewith each amplitude value avoids peaks of excessively high amplitude inthe resulting hologram of the given two-dimensional image.

The step of simulating the production of a diffracted image can includethe computation of at least one of the following complex transforms:Fourier transform, Walsh transform, Hankel transform, orthogonalpolynomial transform, Hadamar transform, Karhunen-Loeve transform,multiresolution discrete wavelet transform, adaptive wavelet transformand a transform consisting of a composite of at least two of the abovetransforms.

The choice of a complex transformation can depend on the distancebetween the first and second geometrical planes, each complextransformation being more particularly suited to a predetermined rangeof distances between the first and second geometrical planes.

To be more precise, the step of simulating the production of adiffracted image can consist of computing a convolutional product,associated with the oversampled complex image, of two components, byapplying the transform which is the inverse of said complex transform tothe product of the respective complex transforms of said two components.

Until now, the skilled person has regarded the Fourier transform, whichis widely used in optics, as the best possible transform for calculatinga convolutional product of this kind. However, experiments conducted bythe present inventors have shown that, for a given two-dimensionalimage, using one of the complex transforms mentioned above other thanthe Fourier transform produces a resultant hologram of much betterquality, i.e. which, when it is reproduced physically and illuminated bya coherent source, produces an image associated with the giventwo-dimensional image that is finer than those generally produced byprior art systems.

The step of combining the holograms can in particular consist ofjuxtaposing the holograms of the two-dimensional images in a commondigital image constituting the hologram of the object.

The present invention also provides a method of producing athree-dimensional image from a virtual object defined in athree-dimensional geometrical space, characterized in that it includesthe following steps:

producing a hologram of the object by the method defined above,

physically reproducing the hologram of the object on a spatial lightmodulator, and

illuminating the spatial light modulator in order to reproduce athree-dimensional image of the object from the hologram.

According to one aspect of the invention, the step of illuminating thespatial light modulator consists of illuminating it with three opticalwaves in turn respectively representing the colors red, green and bluein synchronism with reproduction by the spatial light modulator of asequence of holograms of the object produced by the hologram productionmethod defined above and each corresponding to one of the three colors,so that a three-dimensional color image of the object is reproduced.

A sequence of holograms can also be physically reproduced by the spatiallight modulator, with each of the holograms of the sequence beingobtained by the hologram production method defined above, so that afterthe step of illuminating the spatial light modulator, animatedthree-dimensional images of the object can be reproduced.

The present invention also provides a system for producing a hologramfrom a virtual object defined in a three-dimensional geometrical space,characterized in that it includes:

means for storing in memory the virtual object defined in thethree-dimensional geometrical space,

first computing means for producing a set of two-dimensional imagesrepresenting the object from respective different viewpoints in thethree-dimensional space,

second computing means for producing holograms respectivelycorresponding to the two-dimensional images, and

combining means for combining said holograms to form a hologram of theobject.

The first computing means can comprise projection computing means forcomputing the projection of images of the object as seen from respectivepoints of a matrix of points in a first geometrical plane separate fromthe object onto a second geometrical plane which is preferably betweenthe object and the first plane and parallel to the first plane in thethree-dimensional geometrical space.

The projection computing means can comprise means for computing, foreach point of the matrix, the projection of points of the object ontothe second plane along respective straight lines passing through saidpoints of the object and said each point of the matrix.

The second computing means advantageously comprise:

transform means for transforming a given two-dimensional image definedby a real function into a complex image defined by a complex function,

means for oversampling the complex image,

simulator means for simulating the production of a diffracted imageresulting from the diffraction of an optical wave by the oversampledcomplex image,

means for adding a complex field representing a reference optical waveto the resulting diffractive image, and

means for encoding values taken by the amplitude of the sum of saidcomplex field and the diffracted image to produce the hologramassociated with the given two-dimensional image.

The transform means can comprise:

means for determining amplitude values each depending on the square rootof a corresponding value taken by the real function, and

means for associating a phase with each of the amplitude values so thatan amplitude value and a phase value are defined for each point of thecomplex image.

The simulator means can comprise means for computing one of thefollowing complex transforms: Fourier transform, Walsh transform, Hankeltransform, orthogonal polynomial transform, Hadamar transform,Karhunen-Loeve transform, multiresolution discrete wavelet transform,and adaptive wavelet transform.

More specifically, the simulator means can comprise means for computinga convolutional product, associated with the oversampled complex image,of two components, by applying the transform which is the inverse ofsaid complex transform to the product of the respective complextransforms of said two components.

The combining means can comprise means for juxtaposing the holograms ofthe two-dimensional images in one digital image constituting thehologram of the object.

The present invention also provides a system for producing athree-dimensional image from a virtual object defined in athree-dimensional geometrical space, characterized in that it comprises:

a system as defined above for producing a hologram of the object,

a spatial light modulator for physically reproducing the hologram of theobject, and

a light source for illuminating the spatial light modulator in order toreproduce a three-dimensional image of the object from the hologram.

The spatial light modulator comprises a liquid crystal screen having apixel pitch less than 10 μm and preferably from 1 μm to 2 μm in at leasttwo distinct directions. The expression “pixel pitch” means the periodof reproduction of the pixels in a given direction, which correspondsfor each pixel to the sum of the dimension of the pixel in the givendirection and the distance between that pixel and an adjacent pixel inthe same direction. The distance between two pixels is made as small aspossible and is preferably substantially zero. The aforementioned twodistinct directions respectively correspond to rows and columns ofpixels on the liquid crystal screen.

Said system for producing a hologram of the object, the spatial lightmodulator and the light source can be on the same site. Instead, thesystem for producing a hologram of the object can be on a first site andthe spatial light modulator and the light source on a second site, thefirst and second sites being remote from each other.

Other advantages of the present invention will become apparent onreading the following detailed description with reference to theaccompanying drawings, in which:

FIG. 1 is a block diagram of a first embodiment of a holographic systemaccording to the invention,

FIG. 2 is a diagram of the structure of a spatial light modulator (SLM)used in the system shown in FIG. 1,

FIG. 3 is a flowchart of an algorithm used in the system shown in FIG.1,

FIG. 4 is a diagram showing projection of images by the algorithm shownin FIG. 3,

FIG. 5 is a flowchart of an algorithm used in the system shown in FIG. 1to produce a hologram from a two-dimensional image,

FIG. 6 is a diagram showing oversampling of a two-dimensional image bythe algorithm shown in FIG. 5,

FIG. 7 shows the production of a hologram from a two-dimensional image,

FIG. 8 is a diagram showing a digital hologram generated by thealgorithm shown in FIG. 3, and

FIG. 9 is a block diagram of a second embodiment of a holographic systemaccording to the invention.

Referring to FIG. 1, a first embodiment of a holographic system of theinvention includes a computer 1, a spatial light modulator 2, a driverinterface 3 enabling the spatial light modulator 2 to be controlledbased on signals output by the computer 1, and a light source 4.

The computer 1 contains a virtual three-dimensional object in a memory(not shown). The virtual object is defined, for example, by a set oftriplets of co-ordinates in a system of axes in three-dimensional space,each triplet of co-ordinates corresponding to a point on the externalsurface of the object. The virtual object is typically obtained in theconventional way by a computer-assisted design (CAD) technique or by anyother method of synthesizing voluminal images, such as tomography, or aradiosity or ray launching method associated with a three-dimensionalgeometrical modeling system.

The computer 1 also contains in a memory an algorithm for generatingholograms which is described in more detail below, with reference toFIGS. 3 to 5, and which is used to produce a digital hologramcorresponding to the virtual object. The computer 1 controls the spatialmodulator 2 via the driver interface 3 so that the hologram generateddigitally by the computer 1 is reproduced physically by the spatialmodulator 2.

The light source 4 is a source capable of emitting coherent light at apredetermined wavelength λ, such as a laser or a source of white lightassociated with a dichroic or interference filter. The coherent lengthof the light source 4 is predefined as a function of characteristicdimensions of the spatial light modulator 2, in a manner that is knownto the skilled person. In the embodiment shown in FIG. 1, the lightsource 4 is arranged to illuminate the spatial modulator 2 bytransmission.

The spatial modulator 2 comprises an array of diffractive cells whichare controlled to reproduce the aforementioned digital hologramphysically and which diffract the light emitted by the light source 4 sothat, by virtue of holographic reconstruction, an observer 5 in front ofthe spatial modulator 2 sees a three-dimensional image of the virtualobject. The spatial light modulator 2, also referred to as a“holographic screen” in the specific application to holography, istypically a liquid crystal screen whose states are transparent or opaqueand in which the pixel pitch p in the horizontal and vertical directionsis less than 10 μm and preferably from 1 μm to 2 μm. The pixel pitch pin the horizontal or vertical direction is defined as the sum of thedimension of a pixel in that direction and the distance between thatpixel and an adjacent pixel in the same direction.

FIG. 2 is a diagram showing the structure of a liquid crystal screenused as the spatial modulator 2 in the present invention. The liquidcrystal screen 2 is divided into a predetermined integer number N×M ofelementary screens 20 ₁₁ to 20 _(NM) arranged in a matrix, eachelementary screen comprising a predetermined integer number K×Q ofpixels, also referred to as elementary diffractive cells. The dimensionsof each elementary screen are typically 1 mm×1 mm. In the embodimentshown in FIG. 2, the distance between two adjacent pixels is virtuallyzero and the pixel pitch p is therefore equal to the length of thepixels in the horizontal or vertical direction. The benefit of thisscreen structure will become more apparent on reading the remainder ofthe description.

FIGS. 3 to 5 show the method according to the invention. In particular,FIG. 3 shows an algorithm partially implemented in the computer 1 fromFIG. 1.

In a preliminary step E0 of the algorithm, a three-dimensional virtualobject or scene 6 is stored in a memory of the computer 1. As previouslyexplained, the virtual object 6 is defined by the co-ordinates in athree-dimensional system of axes (O,x,y,z) of points 60 constituting itsexternal surface.

In a first step E1, a first geometrical plane 7, referred to as thehologram computation plane, is defined. In the three-dimensional systemof axes (O,x,y,z) the first plane 7 is at a non-zero distance D1 fromthe virtual object 6, so that the plane 7 and the object 6 arecompletely separate, as shown in FIG. 4.

In a next step E2, part of the first plane 7 is sampled to define amatrix made up of N×M regularly distributed sampling points 70 ₁₁ to 70_(NM). Each sampling point 70 _(nm), where n and m are integersrespectively from 1 to N and from 1 to M, sees the virtual object 6 froma respective viewpoint. The image of the object 6 seen from a givensampling point 70 _(nm) is inscribed within a cone 71 _(nm) whose apexis the given point 70 _(nm), whose generatrices are the half-lines 72_(nm) which originate from the given point 70 _(nm) and rest on theapparent contour 73 _(nm) of the object 6 as seen from the given point70 _(nm), and whose base is the surface 74 _(nm) delimited by theapparent contour 73 _(nm). To simplify the drawing, FIG. 4 shows onlyone cone 71 _(nm).

In a step E3 of the algorithm a second plane 8, referred to as theprojection plane, is defined in the three-dimensional system of axes(O,x,y,z). The second plane 8 is separate from the first plane 7 and ispreferably between the virtual object 6 and the first plane 7, parallelto the first plane 7 and at a non-zero distance D2 from the first plane7. The distance D2 between the planes 7 and 8 corresponds in practice tothe distance from the spatial modulator 2 at which the three-dimensionalimage of the object is reproduced and perceived by the observer 5 (seeFIG. 1).

In a step E4, the image of the virtual object 6, as seen from eachsampling point 70 _(nm), is projected onto the second plane 8 using aconical projection so that the resultant two-dimensional projected image80 _(nm) is inscribed within the cone 71 _(nm), as shown in FIG. 4. Tobe more precise, for each sampling point 70 _(nm) on the plane 7, eachpoint 60 of the outside surface of the virtual object 6 visible from thesampling point 70 _(nm) is projected onto the second plane 8 along thestraight line passing through the point 60 and the sampling point 70_(nm) and in the direction of the sampling point 70 _(nm). Thetwo-dimensional projected image 80 _(nm) is defined digitally by anintensity distribution f_(nm)(Y,Z), in other words, each point (pixel)of the image 80 _(nm), identified by its co-ordinates (Y,Z) in theprojection plane 8, is associated with an intensity value which is areal number.

When the N×M projected images 80 ₁₁ to 80 _(NM) respectivelycorresponding to the N×M sampling points 70 ₁₁ to 70 _(NM) have beendetermined in the second plane 8, holograms 75 ₁₁ to 75 _(NM), one ofwhich is shown diagrammatically in FIGS. 4 and 7, are produced digitallyfor the respective projected images in a next step E5. The hologram 75_(nm) for a given two-dimensional projected image 80 _(nm) can becomputed using a technique known in the art and based on the Fouriertransform. A description of a technique of this kind can be found in thearticle by S. Michelin, D. Arquès and J. C. Grossetie entitled“Fourier-transform computer generated hologram: a variation on theoff-axis principle” published in SPIE Conferences 1994, PracticalHolography VIII, pages 249-254, or in the article by Olof Bryngdahl andFranck Wyrowski, working under E. Wolf, entitled “DigitalHolography-Computer-Generated Holograms”, published in Progress inOptics, Volume XXVIII, by Elsevier Science Publisher B.V., 1990.Generally speaking, this technique simulates the analog production of ahologram by applying a series of Fourier transforms to a convolutionalproduct associated with a two-dimensional image, adding a complex fieldrepresenting a reference optical wave to the series of Fouriertransforms thus obtained, and then extracting the amplitude informationcontained in the sum of the complex field and the series of Fouriertransforms.

The present invention employs an algorithm representing an improvementover the conventional methods to implement step E5. FIG. 5 shows thisalgorithm.

In a step E50, the projected two-dimensional image 80 _(nm) which isdescribed by the aforementioned intensity distribution f_(nm)(Y,Z) istransformed into a transformed two-dimensional image 81 _(nm) defined byan amplitude distribution, by computing for each point of the image 80_(nm) the square root of the corresponding intensity value.

In a next step E51, a “pseudorandom” diffuser is generated digitally.This diffuser consists of an “image” having the same number of pixels asthe projected two-dimensional image 80 _(nm) and in which each pixel hasan intensity value equal to 1 and a random phase. Each phase of thediffuser is then associated with a corresponding pixel of thetransformed two-dimensional image 81 _(nm), to transform the image 81_(nm) into a “complex” image 82 _(nm) in which a complex number definedby an amplitude value and a phase value is determined for each pixel.The pseudorandom diffuser prevents the resulting hologram 75 _(nm)having excessive amplitude level disparities, by averaging the amplitudevalues of the hologram.

In a step E52 the complex image 82 _(nm) obtained in step E51 isoversampled, i.e. the image is included in a larger image, as shown inFIG. 6. An image 83 _(nm) is thus formed consisting of the complex image82 _(nm) in a central part 830 _(nm) and of pixels whose amplitude ischosen arbitrarily, for example equal to 0, in a complementaryperipheral part 831 _(nm). This oversampling of the complex image 82_(nm) increases the number of pixels of the resultant hologram 75 _(nm)and therefore reproduces a three-dimensional image of the object 6 withgreater resolution.

In a step E53, the diffracted image produced in the hologram computationplane 7 when the projected two-dimensional image 80 _(nm) is illuminatedby a fictitious coherent wave DIF of wavelength λ (see FIG. 4) issimulated digitally. Step E53 consists of computing a convolutionalproduct associated with the oversampled complex image 83 _(nm). Theconvolutional product conforms to scalar diffraction theory. Forexample, using a Rayleigh-Sommerfeld scalar diffraction formulation, thetwo components of the convolutional product can respectively correspondto a complex field representing the oversampled complex image 83 _(nm)and a complex field representing a spherical optical wave of wavelengthλ. The skilled person however knows other types of convolutional productfor computing a diffracted image. The convolutional product computed instep E53 uses parameters including the aforementioned distance D2 andthe emission wavelength λ of the light source 4.

In accordance with the invention, the convolutional product is computedby applying a complex transform, also referred to as a fast complextransform, to the two components of the convolutional product, computingthe product of the resulting fast complex transforms, and then applyingthe fast complex transform which is the inverse of said fast complextransform to the aforementioned product of the fast complex transforms.

To be more precise, if CONV denotes the convolutional product, C1 and C2its two components, and T the fast complex transform, then theconvolutional product is written:CONV=C1{circle around (x)}C2=T ⁻¹ T(C1{circle around (x)}C2)CONV=T ⁻¹(T(C1)T(C2)).

In the present context, the expression “fast complex transform” means amathematical transform compatible with scalar optical diffractiontheory, i.e. whose resulting transformed functions satisfy theconventional scalar diffraction equations. The fast complex transformmust also have the property whereby the fast complex transform of aconvolutional product of two components is equal to the product of therespective fast complex transforms of each of said two components. TheFourier transform, the orthogonal polynomial transform, the Paleytransform, the Hadamar transform, the Walsh transform, the Hankeltransform, the Karhunen-Loeve transform, the multiresolution discretewavelet transform and the adaptive wavelet transform are all fastcomplex transforms which meet the above conditions. Other appropriatefast complex transforms are composites of at least two of theaforementioned transforms, such as a composite of the Walsh transformand the Hadamar transform. The application of a composite of twotransforms T1 and T2 to any image I is defined in standard mathematicterms by the equation:(T1∘T2)(I)=T1(T2(I)).

Each of the aforementioned fast complex transforms can be used in aspecific case. In particular, the fast complex transform is chosenaccording to the distance D2 from the spatial light modulator 2 at whichthe three-dimensional optical image of the object 6 is to be reproduced.A Fourier transform is appropriate for a large distance D2. A Walshtransform is more suitable for a smaller distance D2. Also, it has beenfound that using one of the above-mentioned fast complex transformsother than the Fourier transform gives better results in terms of thequality of the hologram 75 _(nm) than those obtained using the Fouriertransform.

It should be noted that, because the projected two-dimensional image 80_(nm) is transformed into a complex image 82 _(nm), computing theconvolutional product associated with the image 80 _(nm) in step E53 ismore practical than in the prior art since the fast complex transform isapplied directly to an image 83 _(nm) defined by a complex function andnot to an image defined by a real function.

At the end of step E53, the diffracted image 84 _(nm) is defined by acomplex field made up of a set of complex numbers each of which isassociated with a point of the image 84 _(nm). Each of these complexnumbers also depends on the image 83 _(nm) taken as a whole.

In a next step E54 a complex field simulating a reference optical waveREF of wavelength λ directed towards the hologram computation plane 7 isadded, in the plane 7, to the complex field representing the diffractedimage 84 _(nm). The amplitude information contained in the resultingcomplex field is then extracted in order to produce an interferencefield. The addition of the aforementioned two complex fields isperformed by adding, at each point of the diffracted image 84 _(nm), thecomplex number associated with that point and the value at the samepoint of the complex field representing the reference wave REF. Theinterference field constitutes the hologram 75 _(nm) of thetwo-dimensional projected image 80 _(nm).

A variant of the FIG. 5 algorithm dispenses with the steps E50 and E51of producing the complex image and/or the oversampling step E52. Inanother variant, the oversampling step E52 precedes the steps E50 andE51 of producing the complex image.

The hologram 75 _(nm) of a given two-dimensional image 80 _(nm) obtainedin step E5 is a diffractive field, or grating, which is computed for aparticular wavelength, namely the emission wavelength λ of the lightsource 4. This hologram, which is present in virtual form in step E5,i.e. represented in the computer 1 by digital data, is such that, if itis reproduced physically by a holographic screen, illuminating saidholographic screen with a laser source emitting at the aforementionedwavelength λ reproduces the original two-dimensional image 80 _(nm) at agiven order of diffraction.

Each hologram 75 _(nm) obtained in step E5 is defined digitally in thecomputer 1 by a two-dimensional amplitude function A_(nm)(u,v), where(u,v) designate co-ordinates in the hologram computation plane 7 whichcorrespond, for example, to image spatial frequencies when the fastcomplex transform chosen in step E53 is a Fourier transform. Thetwo-dimensional amplitude function A_(nm)(u,v) is deduced from thetwo-dimensional intensity function f_(nm)(Y,Z) defining thecorresponding projected two-dimensional image 80 _(nm), as explainedabove. In practice, the function A_(nm)(u,v) associated with a givenhologram 75 _(nm) is computed only for a series of discrete points(u,v)=(u^(k) _(nm), v^(q) _(nm)), where k and q are integersrespectively from 1 to K and from 1 to Q (see FIG. 7). The values thatthe function A_(nm)(u,v) takes can nevertheless be spread continuouslybetween a minimum amplitude value and a maximum amplitude value.

Referring again to FIG. 3, in a step E6 of the algorithm, the valuestaken by the function A_(nm)(u,v) are quantized and encoded, i.e. eachvalue of the function is associated with a discrete value which isencoded digitally, for example on eight bits. To each pair of discretepoints (u^(k) _(nm), v^(q) _(nm)) there then corresponds a discreteamplitude value representing one of 256 gray levels. The amplitudesA_(nm)(u,v) can also be quantized more simply by allocating to eachamplitude value of A_(nm)(u,v) the discrete value “0” if said amplitudevalue is below a predetermined threshold or the discrete value “1” ifsaid amplitude value is above the predetermined threshold.

In a next step E7 the quantized and encoded holograms 90 ₁₁ to 90 _(NM)from step E6 are juxtaposed to form a digital image 9 shown in FIG. 8.The holograms 90 ₁₁ to 90 _(NM) are arranged in the digital image 9 withthe same configuration as the corresponding sampling points 70 ₁₁ to 70_(NM) in the first plane 7. The digital image 9 therefore represents ahologram of the virtual object 6.

In a step E8, the digital image (hologram) 9 is sent by the computer 1to the spatial light modulator 2 via the driver interface 3 so that itcan be physically reproduced by the spatial light modulator 2. To bemore precise, each elementary screen 20 _(nm) displays the correspondinghologram 90 _(nm). By illuminating the spatial modulator 2 by means ofthe light source 4, a real or virtual three-dimensional optical image ofthe object 6 can then be reproduced by diffraction of the light emittedfrom the source 4 by the spatial modulator 2.

The above description applies to a number of pixels (diffractive cells)of the digital image 9 equal to the number of pixels of the holographicscreen 2, i.e. equal to N×M×K×Q, and to exactly the same arrangement ofthe pixels in the digital image 9 and on the screen 2, so that thedigital image 9 is perfectly matched to the structure of the screen 2.However, if the number and/or arrangement of the pixels in the image 9and on the screen 2 are different, an adaptation step E78 precedes thethree-dimensional image reproduction step E8 and adapts the digitalimage 9 to the structure of the holographic screen 2.

As already mentioned, the holograms 90 ₁₁ to 90 _(NM) of thetwo-dimensional images 80 ₁₁ to 80 _(NM), and therefore the hologram 9of the object 6, are computed for the emission wavelength of the lightsource 4. The three-dimensional image of the virtual object 6 istherefore reproduced by the screen 2 in the color corresponding to thatwavelength.

FIG. 9 shows a second embodiment of the holographic system according tothe invention. This second embodiment differs from the first embodiment,shown in FIG. 1, in that the light source 4 is replaced by three lightsources 4 a, 4 b and 4 c respectively producing coherent red, green andblue light. The light sources 4 a, 4 b and 4 c are controlled by thecomputer 1 via a dedicated interface (not shown) so that they emit lightin turn and in synchronism with the reproduction by the spatialmodulator 2 of a sequence of holograms computed by the computer 1 andeach corresponding to one of the colors red, green and blue (RGB). Thuseach light source 4 a, 4 b and 4 c illuminates the spatial modulatorwhen a hologram respectively associated with the color red, green orblue is displayed by the spatial light modulator 2. Using thistime-division multiplexing technique, it is possible to reproduce athree-dimensional color image of the virtual object 6.

The invention is not limited to illumination of the spatial modulator 2by transmission. Thus in the embodiments shown in FIGS. 1 and 9 therespective light sources 4 and 4 a, 4 b, 4 c can be on the same side ofthe spatial modulator 2 as the observer 5, so as to diffract lightreflected from the spatial modulator 2.

The spatial light modulator 2 used in the present invention is capableof reproducing holograms in real time. Accordingly, the method describedwith reference to FIGS. 3 to 5 can be used for sequences ofthree-dimensional images to reproduce animated images.

In the embodiment shown in FIGS. 1 and 9, the holographic system is allon one site. However, the holographic system according to the inventioncan instead be divided into two remote systems, namely a first system,referred to as the “transmitter”, implementing the digital steps E0through E7 of the algorithm shown in FIG. 3, i.e. producing the digitalhologram 9 from the virtual object 6, and the second system, referred toas the “receiver”, implementing the subsequent steps E78 and E8. Thehologram 9 produced by the transmitter is transmitted to the receiver inthe form of a digital signal via a transmission medium.

In another variant of the invention, the holographic system is on onesite but receives the virtual object 6 from a remote transmitter.

1. A method for producing a hologram from a virtual object (6) definedin a three-dimensional geometrical space said method comprising thesteps of: computing (E1-E4) a set of two-dimensional images (80 _(nm))representing the object as seen from respective different viewpoints inthe three-dimensional geometrical space, each of said two-dimensionalimages (80 _(nm)) representing the object as seen from one of saiddifferent viewpoints, computing (E5-E6) a set of elementary holograms(90 _(nm)), each of said elementary holograms corresponding to one ofsaid two-dimensional images, and combining (E7) said elementaryholograms (90 _(nm)) in a combined digital image to form a hologram (9)of the object (6), wherein each of said two-dimensional images (80_(nm)) comprises coordinates (Y,Z) and is defined by an intensitydistribution (f_(nm)(Y,Z)) over said coordinates, and wherein said step(E5-E6) of computing the elementary holograms for a giventwo-dimensional image (80 _(nm)) comprises the following steps:converting (E50, E51) the two-dimensional image defined by thecorresponding real function into a complex image defined by a complexfunction, oversampling (E52) the complex image (82 _(nm)), simulatingillumination of the oversampled complex image by an optical wave (DIF)to obtain a diffracted image (84 _(nm)), adding (E54) a complex fieldrepresenting a reference optical wave (REF) to the resulting diffractedimage (84 _(nm)) to produce an interference field, and extracting (E6)amplitude values of the sum of said complex field and the resultingdiffracted image (84 _(nm)) to produce the hologram (90 _(nm))associated with said given two-dimensional image (80 _(nm)), whereinsaid step of computing the set of two-dimensional images includes thefollowing steps: defining a first geometrical plane (7) in thethree-dimensional geometrical space, said first geometrical plane beingseparate from said object, defining (E1-E2) a matrix of points (70_(nm)) in said first geometrical plane (7), each of said pointscorresponding to one of said different viewpoints, defining a secondgeometrical plane (8), said second geometrical plane (8) being parallelto said first geometrical plane and preferably located between theobject (6) and the first geometrical plane (7), and projecting (E3-E4)images of the object as respectively seen from said points (70 _(nm)) ofsaid matrix onto said second geometrical plane (8), wherein theprojected image constitute said two-dimensional images (80 _(nm)).
 2. Amethod according to claim 1, wherein, for each point (70 _(nm)) of thematrix, said projection step consists of projecting points (60) of theobject (6) onto the second plane (8) along respective straight linespassing through said points of the object and said each point of thematrix.
 3. A method according to claim 1, wherein said step (E5-E6) ofcomputing the holograms is implemented using a technique employing aFourier transform.
 4. A method according to claim 1, wherein saidconverting step includes the following steps: determining (E50)amplitudes associated with pixels of the complex image, said amplitudesdepending, for each pixel of said image, on the square root of acorresponding intensity value of said real function of the giventwo-dimensional image defined by said real function, and associating(E51) a phase with each of said amplitudes so that an amplitude and aphase are defined for each point of the complex image.
 5. A methodaccording to claim 1, wherein said step of simulating illumination ofthe oversampled complex image by an optical wave (DIF) comprises thestep of calculating said diffracted image, said diffracted imagecalculating step comprising the steps of: calculating a convolutionproduct of the oversampled complex image and a function describing saidoptical wave, wherein the convolution product is obtained by firstperforming a complex transformation on said oversampled complex imageand said function describing said optical wave respectively and then aninverse complex transformation of the product of the transformed compleximage and the transformed function describing said optical wave toobtain the diffracted image, wherein said inverse complex transformationis the inverse of said complex transformation.
 6. A method according toclaim 5, wherein said complex transformation is at least one of aFourier transform, Walsh transform, Hankel transform, orthogonalpolynomial transform, Hadamar transform, Karhunen-Loeve transform,multiresolution discrete wavelet transform, adaptive wavelet transformand a transform consisting of a composite of at least two of the abovetransforms.
 7. A method according to claim 1, wherein said step (E7) ofcombining the holograms comprises juxtaposing the holograms (90 _(nm))of the two-dimensional images (80 _(nm)) in said combined digital image(9) constituting said hologram (9) of the object (6).
 8. A method ofproducing a three-dimensional image from a virtual object (6) defined ina three-dimensional geometrical space, comprising the following steps:producing a hologram (9) of the object (6) by a method according to oneof claims 1, 2, 3, and 4 to 7, physically reproducing (E8) said hologram(9) of the object (6) on a spatial light modulator (2), and illuminating(E8) the spatial light modulator (2) in order to reproduce athree-dimensional image of the object (6) from the hologram (9).
 9. Amethod according to claim 8, wherein said spatial light modulator (2)comprises a liquid crystal screen having a pixel pitch less than 10 μmand preferably from 1 μm to 2 μm in at least two different directions.10. A method according to claim 8, wherein the step of illuminating thespatial light modulator (2) consists of illuminating said spatial lightmodulator with three optical waves (4 a, 4 b, 4 c) respectivelyrepresenting the colors red, green and blue (RGB) in turn and insynchronism with reproduction by the spatial light modulator (2) of asequence of holograms of the object, each hologram corresponding to oneof the said three colors, so that a three-dimensional color image of theobject (6) is reproduced.
 11. A method according to claim 8, wherein asequence of holograms is physically reproduced by the spatial lightmodulator (2) so as to reproduce animated three-dimensional images ofthe object (6) after the step of illuminating the spatial lightmodulator.
 12. A system for producing a hologram from a virtual object(6) defined in a three-dimensional geometrical space, comprising: memorymeans (1) for storing the virtual object (6) defined in thethree-dimensional geometrical space, first computing means (1) forproducing a set of two-dimensional images (80 _(nm)) representing theobject (6) as seen from respective different viewpoints in thethree-dimensional geometrical space, each of said two-dimensional images(80 _(nm)) representing the object as seen from one of said differentviewpoints; second computing means (1) for producing elementaryholograms (90 _(nm)), each of said elementary holograms corresponding toone of said two-dimensional images (80 _(nm)), and combining means (1)for combining said elementary holograms (90 _(nm)) in a common digitalimage to form a hologram (9) of the object (6), wherein each of saidtwo-dimensional images (80 _(nm)) comprises coordinates (Y,Z) and isdefined by an intensity distribution (f_(nm)(Y,Z)) over saidcoordinates, and wherein the second computing means comprise: convertingmeans (1) for converting (E50, E51) a given two-dimensional image (80_(nm)) defined by the corresponding real function into a complex imagedefined by a complex function, means (1) for oversampling (E52) thecomplex image, means for simulating illumination of the oversampledcomplex image by an optical wave (DIF) to obtain a diffracted image (84_(nm)), means (1) for adding (E54) a complex field representing areference optical wave (REF) to the resulting diffracted image (84_(nm)) to produce an interference field, and means (1) for extracting(E6) values of the amplitude of the sum of said complex field and thediffracted image (84 _(nm)) to produce the hologram (90 _(nm))associated with said given two-dimensional image (80 _(nm)), whereinsaid first computing means comprise projection computing means (1) forcomputing a projection of images of said object (6) as seen fromrespective points (70 _(nm)) of a matrix of points in a firstgeometrical plane (7) separate from the object (6) onto a secondgeometrical plane (8) which is preferably between the object (6) and thefirst plane (7) and parallel to the first plane (7) in thethree-dimensional geometrical space (O, x, y, z), wherein each of saidpoints of said matrix of points corresponds to one of said differentviewpoints.
 13. The system claimed in claim 12, wherein said projectioncomputing means comprise means (1) for computing, for each point (70_(nm)) of the matrix, the projection of points (60) of the object (6)onto the second plane (8) along respective straight lines passingthrough said points of the object and said point of the matrix.
 14. Asystem according to claim 12, wherein said converting means comprise:means (1) for determining (E50) amplitudes associated with each pixel ofsaid complex image, said amplitudes depending, for each pixel of saidimage, on the square root of a corresponding intensity value of saidreal function of the given two-dimensional image defined by said realfunction, and means (1) for associating (E51) a phase with each of saidamplitudes so that an amplitude and a phase are defined for each pointof the complex image.
 15. A system according to claim 12, wherein saidsimulator means comprise means for calculating said diffracted image bycalculating a convolution product of the oversampled complex image and afunction describing said optical wave, wherein the convolution productis obtained by first performing a complex transformation on saidoversampled complex image and said function describing said optical waverespectively and then an inverse complex transformation of the productof the transformed complex image and the transformed function describingsaid optical wave to obtain the diffracted image, wherein said inversecomplex transformation is the inverse of said complex transformation.16. A system according to claim 15, wherein said complex transformationis at least one of a Fourier transform, Walsh transform, Hankeltransform, orthogonal polynomial transform, Hadamar transform,Karhunen-Loeve transform, multiresolution discrete wavelet transform,adaptive wavelet transform and a transform consisting of a composite ofat least two of the above transforms.
 17. A system according to claim12, wherein the combining means (1) comprise means for juxtaposing theholograms (90 _(nm)) of the two-dimensional images (80 _(nm)) in saidcombined digital image (9) constituting said hologram of the object (6).18. A system for producing a three-dimensional image from a virtualobject (6) defined in a three-dimensional geometrical space, comprising:a system according to one of claims 12, 13, and 14 to 17 for producing ahologram (9) of the object (6), a spatial light modulator (2) forphysically implementing the hologram (9) of the object, and a lightsource (4) for illuminating the spatial light modulator (2) in order toreproduce a three-dimensional image of the object (6) from the hologram(9).
 19. A system according to claim 18, wherein said spatial lightmodulator (2) comprises a liquid crystal screen having a pixel pitchless than 10 μm in at least two different directions.
 20. A systemaccording to claim 18, wherein said light source comprises threeseparate light sources (4 a, 4 b, 4 c) for illuminating the spatiallight modulator (2) with three optical waves respectively representingthe colors red, green, and blue (RGB) in turn and in synchronism withthe reproduction by the spatial light modulator (2) of a sequence ofholograms of the object, each hologram corresponding to one of saidthree colors so that a three-dimensional color image of the object isreproduced.
 21. A system according to claim 18, wherein said system forproducing a hologram of said object is on a first site, the spatiallight modulator (2) and the light source (4) are on a second site andthe first and second sites are remote from each other.
 22. A systemaccording to claim 18, wherein said pixel pitch is between 1 μm to 2 μmin at least two different directions.